# Error Propagation - Estimating Variance

1. Oct 29, 2013

### unscientific

1. The problem statement, all variables and given/known data

Not exactly a homework question, but rather a section in Statistical Data Analysis:

Suppose there is a pdf y(x)[/SUB] that is not completely known, but μi and Vij are known:

2. Relevant equations

3. The attempt at a solution

I understand how <y(x)> ≈ y(μ),

My confusion:

Why does <y(x2)>

1. Imply we square everything throughout?

<[y(μ) + Ʃ[∂y/∂x](xi - μi)]2>

2. give a xi and xj term? Where did the xj come from?

3. Why is it for i≠j when xi and xj are uncorrelated, the expression simplifies to

σ2y ≈ Ʃ[∂y/∂x]2σ2i

Where did the j go?

Last edited: Oct 29, 2013
2. Oct 30, 2013

### Ray Vickson

It told you explicitly where the j "went": it said that $V_{ii} = \sigma_i^2$ and that $V_{ij} = 0$ for $i \neq j$.

3. Oct 31, 2013

### unscientific

Hmm, that makes sense.

What about the initial derivation? Why did they choose to square the entire RHS when it's a function of (x2) and not f2(x)? And the j's started appearing..

4. Oct 31, 2013

### Ray Vickson

Do you honestly mean to say that you cannot tell the difference between $g(x)^2$ and $g(x^2)$? The paper is working with $g(x)^2$!

5. Oct 31, 2013

### unscientific

Ah I see, but where did the j's come from though? And nothing about y(x2) was said that day.

6. Oct 31, 2013

### Ray Vickson

Try it for yourself: write out $(\sum_{i=1}^3 a_i)^2 = (a_1 + a_2 + a_3)^2$ in complete detail, by expanding out the square. After doing that, re-write the result using summation notation.

This will be my last post on this topic.